Group #13047986 — subgroups

Subgroups — #13047986

Order 1296.

189 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#737	C₂	1	1	yes	maximal, characteristic
2	#10619965	C₂	1	1	yes	maximal, characteristic
3	#10619724	C₂	1	1	yes	maximal, characteristic
4	#88547	C₃	1	1	yes	maximal, characteristic, complemented
5	#3674⋊C₃	—	3	2	no	maximal
6	#3756⋊C₃	—	3	1	no	maximal
7	C₃×#3674	—	3	1	no	maximal
8	C₄×#737	—	2	1	no	—
9	C₄×#737	C₂²	1	1	yes	characteristic
10	#66409	—	2	1	no	—
11	Q₈×3₋¹⁺²	—	2	1	no	—
12	Q₈×3₋¹⁺²	—	2	2	no	—
13	Q₈×3₋¹⁺²	D₃	1	2	yes	—
14	Q₈×H(3,𝔽₃)	D₃	1	1	yes	characteristic
15	C₃×Q₈×C₉	—	2	1	no	—
16	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
17	C₃×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
18	#6226	C₂×C₃	1	1	yes	characteristic
19	3₋¹⁺²⋊Q₈	—	3	2	no	—
20	#634⋊C₃	—	3	1	no	—
21	C₃×Dic₁₈	—	3	1	no	—
22	C₃⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	3₋¹⁺²⋊C₈	—	3	2	no	—
24	(C₃⋊(C₃⋊C₈))⋊C₃	—	3	1	no	—
25	C₃×C₉⋊C₈	—	3	1	no	—
26	C₂×#737	D₄	1	1	yes	characteristic
27	#3756	—	3	1	no	—
28	C₃×#216	—	9	1	no	—
29	#3674	—	3	1	no	—
30	#3674	—	3	2	no	—
31	C₄×3₋¹⁺²	—	2	2	no	—
32	C₄×3₋¹⁺²	—	2	4	no	—
33	C₄×3₋¹⁺²	—	2	1	no	—
34	C₄×3₋¹⁺²	—	2	2	no	—
35	C₄×3₋¹⁺²	—	2	2	no	—
36	C₄×3₋¹⁺²	C₂×D₃	1	2	yes	—
37	C₄×H(3,𝔽₃)	—	2	1	no	—
38	C₄×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
39	C₃×C₄×C₉	—	2	2	no	—
40	C₃×C₄×C₉	—	2	1	no	—
41	C₃×C₄×C₉	—	2	1	no	—
42	C₃×C₄×C₉	—	2	1	no	—
43	C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
44	C₃×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
45	C₃⋊Dic₉	—	2	1	no	—
46	3₋¹⁺²⋊C₄	—	6	2	no	—
47	(C₃⋊Dic₃)⋊C₃	—	6	1	no	—
48	C₃×Dic₉	—	6	1	no	—
49	#737	Q₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₃²×Q₈	—	2	1	no	—
51	C₃²×Q₈	C₃×D₃	1	1	yes	characteristic
52	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
53	#634	—	3	1	no	—
54	C₃×Dic₆	—	9	1	no	—
55	C₃⋊(C₃⋊C₈)	—	3	1	no	—
56	C₃×C₃⋊C₈	—	9	1	no	—
57	Q₈×C₉	—	2	2	no	—
58	Q₈×C₉	—	2	1	no	—
59	Q₈×C₉	C₃×D₃	1	1	yes	characteristic
60	Q₈×C₉	C₃×D₃	1	2	yes	—
61	Dic₁₈	—	3	1	no	—
62	Dic₁₈	—	3	2	no	—
63	C₉⋊C₈	—	3	1	no	—
64	C₉⋊C₈	—	3	2	no	—
65	C₂×3₋¹⁺²	—	2	1	no	—
66	C₂×3₋¹⁺²	—	2	2	no	—
67	C₂×3₋¹⁺²	#67	1	2	yes	—
68	C₂×H(3,𝔽₃)	#67	1	1	yes	characteristic
69	C₂×C₃×C₉	—	2	1	no	—
70	C₂×C₃×C₉	#67	1	1	yes	characteristic
71	C₂×C₃×C₉	C₃×D₄	1	1	yes	characteristic
72	C₃×Q₁₆	—	27	1	no	—
73	#216	—	9	1	no	—
74	#216	—	27	1	no	—
75	C₃²×C₄	—	2	2	no	—
76	C₃²×C₄	—	2	1	no	—
77	C₃²×C₄	—	2	1	no	—
78	C₃²×C₄	—	2	1	no	—
79	C₃²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
80	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
81	C₃⋊Dic₃	—	6	1	no	—
82	C₃×Dic₃	—	18	1	no	—
83	C₄×C₉	—	2	4	no	—
84	C₄×C₉	—	2	2	no	—
85	C₄×C₉	—	2	1	no	—
86	C₄×C₉	—	2	2	no	—
87	C₄×C₉	—	2	2	no	—
88	C₄×C₉	—	2	1	no	—
89	C₄×C₉	C₂×C₃×D₃	1	1	yes	characteristic
90	C₄×C₉	C₂×C₃×D₃	1	2	yes	—
91	Dic₉	—	6	1	no	—
92	Dic₉	—	6	2	no	—
93	3₋¹⁺²	—	2	1	no	—
94	3₋¹⁺²	—	2	2	no	—
95	3₋¹⁺²	#216	1	2	yes	complemented
96	H(3,𝔽₃)	#216	1	1	yes	characteristic
97	C₃×C₉	—	2	1	no	—
98	C₃×C₉	#216	1	1	yes	characteristic, complemented
99	C₃×C₉	C₃×Q₁₆	1	1	yes	characteristic, complemented
100	C₃×Q₈	—	3	1	no	—
101	C₃×Q₈	—	3	1	no	—
102	C₃×Q₈	—	6	1	no	—
103	C₃×Q₈	—	27	1	no	—
104	C₃×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
105	Dic₆	—	9	1	no	—
106	Dic₆	—	27	1	no	—
107	C₃×C₈	—	27	1	no	—
108	C₃⋊C₈	—	9	1	no	—
109	C₃⋊C₈	—	27	1	no	—
110	C₂×C₃²	—	2	1	no	—
111	C₂×C₃²	C₃×#67	1	1	yes	characteristic
112	C₂×C₃²	#638	1	1	yes	characteristic
113	C₂×C₉	—	2	2	no	—
114	C₂×C₉	—	2	1	no	—
115	C₂×C₉	C₃×#67	1	1	yes	characteristic
116	C₂×C₉	C₃×#67	1	2	yes	—
117	Q₁₆	—	27	1	no	Sylow
118	C₃×C₄	—	2	1	no	—
119	C₃×C₄	—	3	1	no	—
120	C₃×C₄	—	3	1	no	—
121	C₃×C₄	—	6	1	no	—
122	C₃×C₄	—	6	1	no	—
123	C₃×C₄	—	6	2	no	—
124	C₃×C₄	—	6	1	no	—
125	C₃×C₄	—	54	1	no	—
126	C₃×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
127	Dic₃	—	18	1	no	—
128	Dic₃	—	54	1	no	—
129	C₃²	—	2	1	no	—
130	C₃²	C₃×#216	1	1	yes	characteristic
131	C₃²	#3756	1	1	yes	characteristic
132	C₉	—	2	2	no	—
133	C₉	—	2	1	no	—
134	C₉	C₃×#216	1	1	yes	characteristic
135	C₉	C₃×#216	1	2	yes	—
136	Q₈	—	27	1	no	—
137	Q₈	#4207	1	1	yes	characteristic
138	C₈	—	27	1	no	—
139	C₂×C₃	—	3	1	no	—
140	C₂×C₃	—	3	1	no	—
141	C₂×C₃	—	6	1	no	—
142	C₂×C₃	C₃×#638	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	54	1	no	—
145	C₄	C₂×#4207	1	1	yes	characteristic
146	C₃	—	3	1	no	—
147	C₃	—	3	1	no	—
148	C₃	—	6	1	no	—
149	C₃	C₃×#3756	1	1	yes	characteristic
150	C₂	#10619983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#737

C₂

yes

maximal, characteristic

#10619965

C₂

yes

maximal, characteristic

#10619724

C₂

yes

maximal, characteristic

#88547

C₃

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

#66409

—

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

D₃

yes

—

Q₈×H(3,𝔽₃)

D₃

yes

characteristic

C₃×Q₈×C₉

—

C₃×Q₈×C₉

D₃

yes

characteristic

C₃×Q₈×C₉

C₂×C₃

yes

characteristic

#6226

C₂×C₃

yes

characteristic

3₋¹⁺²⋊Q₈

—

#634⋊C₃

—

C₃×Dic₁₈

—

C₃⋊(C₉⋊C₈)

C₂×C₃

yes

characteristic

3₋¹⁺²⋊C₈

—

(C₃⋊(C₃⋊C₈))⋊C₃

—

C₃×C₉⋊C₈

—

C₂×#737

D₄

yes

—

—

—

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×D₃

yes

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₃

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×D₃

yes

characteristic

C₃×C₄×C₉

C₂²×C₃

yes

characteristic

C₃⋊Dic₉

—

3₋¹⁺²⋊C₄

—

(C₃⋊Dic₃)⋊C₃

—

C₃×Dic₉

—

#737

Q₁₆

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

C₃×D₃

yes

characteristic

C₃²×Q₈

C₃⋊D₃

yes

characteristic

#634

—

C₃×Dic₆

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

C₃×D₃

yes

characteristic

Q₈×C₉

C₃×D₃

yes

—

Dic₁₈

—

Dic₁₈

—

C₉⋊C₈

—

C₉⋊C₈

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#67

yes

—

C₂×H(3,𝔽₃)

#67

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

#67

yes

characteristic

C₂×C₃×C₉

C₃×D₄

yes

characteristic

C₃×Q₁₆

—

#216

—

#216

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×C₃×D₃

yes

characteristic

C₃²×C₄

C₂×C₃⋊D₃

yes

characteristic

C₃⋊Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₃×D₃

yes

characteristic

C₄×C₉

C₂×C₃×D₃

yes

—

Dic₉

—

Dic₉

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

yes

yes

C₃×C₉

—

C₃×C₉

#216

yes

characteristic, complemented

C₃×C₉

C₃×Q₁₆

yes

characteristic, complemented

100

C₃×Q₈

—

101

C₃×Q₈

—

102

C₃×Q₈

—

103

C₃×Q₈

—

104

C₃×Q₈

C₃×C₃⋊D₃

yes

characteristic

105

Dic₆

—

106

Dic₆

—

107

C₃×C₈

—

108

C₃⋊C₈

—

109

C₃⋊C₈

—

110

C₂×C₃²

—

111

C₂×C₃²

C₃×#67

yes

characteristic

112

C₂×C₃²

#638

yes

characteristic

113

C₂×C₉

—

114

C₂×C₉

—

115

C₂×C₉

C₃×#67

yes

characteristic

116

C₂×C₉

C₃×#67

yes

—

117

Q₁₆

—

Sylow

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

C₂×C₃×C₃⋊D₃

yes

characteristic

127

Dic₃

—

128

Dic₃

—

129

C₃²

—

130

C₃²

C₃×#216

yes

characteristic

131

C₃²

#3756

yes

characteristic

132

C₉

—

133

C₉

—

134

C₉

C₃×#216

yes

characteristic

135

C₉

C₃×#216

yes

—

136

Q₈

—

137

Q₈

#4207

yes

characteristic

138

C₈

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

C₃×#638

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

C₂×#4207

yes

characteristic

146

C₃

—

147

C₃

—

148

C₃

—

149

C₃

C₃×#3756

yes

characteristic

150

C₂

#10619983

yes

characteristic

	subgroup	G/H
center	C₂	#10619983
commutator, cosocle	C₃×C₄×C₉	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×C₃⋊D₃
Fitting	Q₈×#737	C₂
socle	C₂×C₃	C₃×#638
prime radical	C₂×H(3,𝔽₃)	#67
hypercenter	Q₈	#4207
hypocenter	C₃×C₉	C₃×Q₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

subgroup

G/H

center

C₂

#10619983

commutator, cosocle

C₃×C₄×C₉

C₂²×C₃

Frattini

C₃×C₄

C₂×C₃×C₃⋊D₃

C₂

C₂×C₃

C₂×H(3,𝔽₃)

Q₈

C₃×C₉

C₃×Q₁₆

LCS

G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₈

N	H ≅ G/N	type
4. #88547	C₃	semidirect
49. #737	Q₁₆	semidirect
95. 3₋¹⁺²	#216	semidirect
98. C₃×C₉	#216	semidirect
99. C₃×C₉	C₃×Q₁₆	semidirect

H ≅ G/N

type

4. #88547

C₃

semidirect

49. #737

Q₁₆

semidirect

95. 3₋¹⁺²

#216

semidirect

98. C₃×C₉

#216

semidirect

99. C₃×C₉

C₃×Q₁₆

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×#737	2
#10619965	2
#10619724	2
#88547	3
#3674⋊C₃	3
#3756⋊C₃	3
C₃×#3674	3

subgroup

index

Subgroups — #13047986

Conjugacy classes of subgroups

Complements

Maximal subgroups